This seems an innocent inequality.
The trivial case when $a \ge 1$ or $b \ge 1$ is easy to prove:
if $a \ge 1$, then $a^b \ge a^0 = 1$.
So the hard part of the problem is when $a, b \in (0, 1)$
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Neat question. Since you've reduced to the case that $a,b\in(0,1)$, perhaps try assuming WLOG that $a\geq b$ and seeing if that helps? – Aweygan Dec 26 '16 at 02:20
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3Possible duplicate of this, this, this or this – Sil Dec 26 '16 at 02:28
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Or this and this – Sil Dec 26 '16 at 02:30